Why did northerners and southerners care about the balance of slave states and free states?
People in both the North and the South cared about the balance between slave and free states because a change in that balance would give one side or the other the majority of power in the Senate. In the US Senate, each state has two senators, regardless of how many people live in a given state. So long as the number of slave and free states was equal, each side had the same number of votes in the Senate. If the balance were to be upset, one side would have more power.
This was important because the two regions had very different interests. They had very different kinds of economies and societies so they disagreed on many things. Therefore, neither side wanted the other to be able to have more power. This is why neither side wanted the other to gain more states and, thereby, more senators.